=Paper=
{{Paper
|id=Vol-2498/short9
|storemode=property
|title=Delta range positioning assisted with satellites
|pdfUrl=https://ceur-ws.org/Vol-2498/short9.pdf
|volume=Vol-2498
|authors=Alexandre Vervisch,Nel Samama
|dblpUrl=https://dblp.org/rec/conf/ipin/VervischS19
}}
==Delta range positioning assisted with satellites==
https://ceur-ws.org/Vol-2498/short9.pdf
Delta Range Positioning Assisted With Satellites
Vervisch-Picois Alexandre 1and Samama Nel1
1
SAMOVAR-Télécom SudParis
Institut Polytechniques de Paris Evry, France
alexandre.vervisch-picois@telecom-sudparis.eu
Abstract. We know that real global positioning, i.e. in all environments, has
been a scientific challenge for several years. The Global Positioning System
(GPS) launched the impulse with its democratization in the 2000s: geographical
position has gradually become a constituent element of modern digital devices
available to general public. Integrating a GPS chip into smartphones was only
one step. The rise of the Internet Of Thing announced by the deployment of 5G
reinforces this trend. Indeed, more and more connected objects will be present
and many of their applications need the geographical position. In this context,
satellite signals play a key role in calculating position, especially outdoors using
the GPS function. In more challenging environments, or when the need for ac-
curacy becomes crucial, these signals can be used alone or in conjunction with
others (or other physical measurements). The recent deployment of several con-
stellations of what are now called the Global Navigation Satellites Systems
(GNSS) further reinforces the presence of these signals, which can therefore be
used. The previous works [1-2] focused on a positioning method using the
measurements of variations of distances between a mobile receiver device and
transmitters (pseudolites) placed in the immediate environment. The same prin-
ciples applied with GNSS satellites give the displacement vectors with a good
accuracy. This allows carrying out positioning assisted with satellite whose re-
sults are presented here.
Keywords: Least-Square algorithm, GNSS, Carrier Phase measurements;
Pseudolites.
1 Delta Range Positioning
1.1 Introduction
We know that positioning with GNSSs has an accuracy of a few meters outdoor (typi-
cally 7 to 12 meters User Range Error for GPS [3]). The principle of GNSS position-
ing is based on signal delay measurements between satellites and receiver. To meas-
ure these propagation times, the GPS receiver uses the "code" component of the
GNSS signal [4]. This component allows an accuracy of a few meters on the meas-
urements of satellite to receiver distances. To improve this accuracy, the carrier phase
of the signal can be used. This one is indeed less sensitive to errors. We can obtain a
2
measurement with an accuracy of a few cm, even a few mm under certain conditions,
for instance for geodesic applications [5]. However, its use is problematic. Indeed, the
receiver can only measure a fraction of the carrier phase whose maximum value is
equal to , the wavelength of the signal (for example = 19 cm for the GPS signal on
L1 band). Thus, of the total distance between a satellite and the receiver, only one
fragment is known. The remainder is a multiple of the wavelength. The distance can
be expressed as in (1):
(1)
Subsequent paragraphs, however, are indented. N is what is called the integer am-
biguity. The calculation of precise positioning with satellites can often be reduced to
the search for the values of the integer ambiguities. There are many methods that are
based on the search for these ambiguities. Perhaps the best known is the phase differ-
ential GNSS, whose most famous application is better known as Real Time Kinematic
(RTK). This is a Differential GNSS method that requires a base station whose position
is known. The pseudoranges (or distances measurements) measured with code and
carrier phase are double differenced to form a new set of equation that allows the de-
termination of ambiguities, with determination of a minimum of a cost function [6].
Other older methods exist such as the triple difference. The triple difference intro-
duces a notion of time to solve the ambiguities. The double differences are taken at two
different moments and differenced once again [7]. Actually, the movements of the
satellites are used to solve the ambiguities. This necessitates waiting several minutes
before getting a position [8].
The Delta Range method, which we will return to in the next section, bypasses the
problem of complete ambiguity by focusing on variations in distances as the receiver
moves. In theory, we no longer need to know the value of the transmitter/receiver
distance: its variation is sufficient.
1.2 Delta Range Positioning Method
The proposed method was based on the measurement of the variation of the distance
“Pseudolite-Receiver” between two positions. In this section, we present the equation
system to be solved, which derives from the basic equations. We present here a sim-
plified version.
Let’s consider that we have eight beacons (there could be more or less, but to ease
the demonstration we use eight). These beacons are transmitting towards (or receiving
from, depending of the technology) a device which wants to know its position in the
area. Fig.1. shows the typical positing situation. Measurements are carried out be-
tween two instants t2 and t1, we could consider more successive instants [1], but for
the sake of clarity, we will limit ourselves to two. The variation of the distances “bea-
cons-device” between t2 and t1 are measured. Equation (2) gives the mathematical
relationship between the unknowns of the issue and these measurements:
(2)
3
With:
(3)
k = {1..8} refers to the kth beacon.
(xj,yj,zj) the coordinates of the position device at the instant tj
the coordinates of the transmitters at the instant tj
the measured difference of distance “beacon k-device” between the instant 2
and the instant 1.This measurement includes any error .
Fig. 1. Principle of positioning with a local constellation of pseudolites.
For 3D positioning, we thus have 6 unknowns (x1; y1; x2; y2; z1; z2). The coordi-
nates of the beacons are fixed and considered known. With 8 beacons, we obtain 8
independent equations which can theoretically solve the issue. As explained in [4], we
can add equations by making measurements at a third instant t3, but this is not neces-
sary here. We will then have asset of 6 unknowns {x1; y1; z1; x2; y2; z2} and 8 equa-
tions (with the ). This corresponds to an over determined systems of equations.
The next step consists of linearization of the previous equations. We note as a
function of the unknown:
(4)
This function is developed in a first order Taylor series about the approximate
“position”. The approximate position corresponds to the set: .
We note X = {dx1; dy1; dz1; dx2; dy2; dz2}, the variations around the approximate
position. Equation (2) is the Taylor development of f k around the approximate posi-
tion:
(5)
With (X) the second and higher orders terms.
All this is just as in Gauss-Newton GNSS positioning algorithm [4]. However, it is
4
here apply to a difference of Euclidian distances (3) instead of a single distance.
Following equation (6) is obtained:
(6)
With:
(7)
(8)
and (9)
Thus linearized, the system of equations to solve can now be written as a matrix
product:
(10)
With:
(11)
and:
(12)
We use the Least-Square algorithm because the system is overdetermined. The inver-
sion of (10) gives (13):
(13)
t
X gives the variation between the approximate position (which is a hypothetic posi-
tion) and the real position (or more precisely a position that is coherent with the
measurement d). The next step consists of updating :
(14)
Then, we go back to (6) to apply (13) with the new . The system is thus iteratively
solved until the error is equal to zero (in practice and << 0). The purpose of
5
the algorithm is finally to find the which minimizes the set of functions .
Note that we consider that the variation of the clock bias between the transmitters and
the receiver is known. This is not true in practice [1]. We prefer to focus on geomet-
rical issues; the very specific issue of the clock bias will be addressed in future works.
At this stage, we would like to apply the principles of the previously presented Delta
Range positioning with GNSS satellites as transmitters.
2 Delta Range Positioning with Satellites
2.1 Displacement Vector with satellites
It is not possible to apply the Delta Range method directly with satellites. In a nut-
shell, it can be said that the very large distances involved with satellites are responsi-
ble for this. This can be understood by looking at equations (8) and (11). If the dis-
tance is very large with respect to the differences between coordinates (x2,y2,z2)
and (x1,y1,z1), the matrix H will not be invertible. Indeed, in this case, even if the sat-
ellite is moving, i.e. (xbk, ybk, zbk) change between instant 1 and 2, we have: ,
and for each transmitter k. Then H is not invertible.
However, the algorithm can be adapted. We can demonstrate that the large dis-
tances can be compensated by considering approximation taking into account the
satellite motion. We can thus obtain not exactly the positions, but the displacement
vector between the position of the receiver at t1 and the position at t2.
(15)
We will detail the method to obtain this vector in future publications. We are pre-
senting now some simulation results of the determination of the vector for different
scenarios.
2.2 Simulations Results with satellites
The simulation results presented here correspond to a displacement of 1 to a few
meters, starting from a point on the Earth's surface with the following coordinates:
P1: Lat: 48.6198530° N, Lon: 2.430451° E, Alt: 105 m
The displacement remains in two dimensions on the Earth's surface. We use 7 GPS
satellites with real ephemeris. Their distribution around position P 1 corresponds to a
6
PDOP of about 1.5 [4]. The results are tested for 1, 5 and 10 seconds of travel, for
lengths of 1, 2, 5 and 10 meters.
We try to simulate more or less realistic reception conditions outdoors. A Gaussian
error centered with standard deviation of 0.001 m, 0.01 m and 0.1 m is randomly add-
ed to the assumed carrier phase measurements. This is typically the kind of error that
is obtained on a phase measurement for different reception conditions [9].
Tables I, II and III present the results obtained in terms of error on the displacement
vector.
Table I: Simulation Data for an error = 0.001m
Distance P1P2
Duration
1m 2m 5m 10m
1 sec 1.1 mm 1.1 mm 1.1 mm 0.9 mm
5 sec 1.7 mm 1.9 mm 2.2 mm 2.4 mm
10 sec 3 mm 3.3 mm 3.8 mm 5 mm
Table II: Simulation Data for an error = 0.01m
Distance P1P2
Duration
1m 2m 5m 10m
1 sec 7 mm 6.9 mm 6.8 mm 6.7 mm
5 sec 3.8 mm 3.5 mm 2.9 mm 2.8 mm
10 sec 0.3 mm 0.3 mm 1.5 mm 5.4 mm
Table III: Simulation Data for an error = 0.1m
Distance P1P2
Duration
1m 2m 5m 10m
1 sec 60 mm 50 mm 47 mm 46 mm
5 sec 99 mm 73 mm 69 mm 67 mm
10 sec 116 mm 94 mm 97 mm 98 mm
In Table I, we observe a phenomenon of deterioration in performance (albeit small)
linked both to the total distance travelled and also to the duration of the movement.
This is not surprising, it is simply the illustration that the higher order terms become a
little less negligible for larger distance covered and for longer duration of the motion
(which results in a larger satellite displacement). In practical terms, this means that it
would be better to favor short movements over short durations.
Table II does not confirm all these trends. The presence of a higher measurement
noise complicates the analysis a little. For the one-second travel time, the logic is
respected: overall deterioration in performance compared to Table I, but general sta-
bility of the error is observed. For 5 seconds duration, performance improves with
distance. It can be deduced that the effect of second-order terms is superseded by the
7
effect of improvement with respect to noise related to a larger distance. This is a phe-
nomenon already observed with the Deltarange algorithm: the more the distance trav-
elled increases, the less sensitive the algorithm is to noise. This is caused by a phe-
nomenon comparable to that of dilution of precision [1]. For the duration of 10 se-
conds overall, the performance is better than with less noise, except for a distance of
10 meters covered, for which it is comparable.
Table III shows performances ranging from 5 to 12 cm of error depending on the
case. It can be reminded that a 10 cm error on a phase measurement means that we are
under difficult reception conditions. We still manage to obtain a displacement vector
with an error of a few cm. We are quite within the orders of magnitude of reasonable
error for a position calculation.
2.3 Experimentation Results with Satellites assisted Delta Range method
Now we would like to use these results to carry out assistance of Delta Range
method. The main advantage of the knowledge of vector is the reduction of the
number of unknowns. Indeed, the Delta Range algorithm can consider two successive
instants t1 and t2, as we presented here, or more (ti). Then we consider a set of dis-
placement vectors in this form:
(16)
If we note , , the knowledge of leads
to the reduction of the number of unknowns to the coordinates of the initial position
P1 (x1,y1,z1) only. Indeed, if we consider our example, the set of unknowns {x1; y1; z1;
x2; y2; z2} can be written: .
We are illustrating this by using this knowledge as assistance of previous experi-
ments presented in [1], reporting the errors previously obtained with simulations on
vector determination. This consists of deploying four transmitters (pseudolites)
emitting a GPS signal and measuring the phase variation in an urban canyon-type
environment measuring 20 m by 30 m. Table IV indicates the results obtained in
terms of determination of the starting point P 1.
Table IV: Error on P1 determination (x tr = number of transmitters, b = best, w = worst)
Error applied to vector D
Numbers 4 tr 3 tr 2 tr (b) 2 tr (w)
of Points 1.1 mm 6.9 mm 50 mm 1.1mm 1.1mm 1.1mm
2 47 cm 40 cm 54 cm 57 cm 28cm 4.3 m
3 36 cm 39 cm 78 cm 31 cm 24 cm 2.1 m
8
3 Conclusion and Future Works
There are a lot of factors that must be taken into account to explain the results on
Table IV. We notice that globally, increasing the error on vector determination
increases the error on P1. This is coherent. We can decrease the number of transmit-
ters, but the quality of the measurements will have a bigger influence than with sever-
al transmitters. The last two columns of Table IV show the worst and the best results
obtained. These results largely depend of the quality of the measured differences of
distance for the two considered transmitters.
Our future works consist of carry out experiment of vector determination and in-
tegrating the assistance to local transmitters positioning method.
References
1. A. Vervisch-Picois, N. Samama, “Delta Carrier Range Algorithm for Indoor Positioning,”
IEEE International Conference on Indoor Positioning and Indoor Navigation, Oct. 2016,
Alcala de Henares, Spain.
2. A. Vervisch-Picois, N. Samama, “Specific Study of Delta Range Positioning,” IEEE Inter-
national Conference on Indoor Positioning and Indoor Navigation, Sept. 2018, Nantes,
France.
3. Brent A. Renfro, Miquela Stein, Nicholas Boeker, Emery Reed and Eduardo Villalba, “An
Analysis of Global PositioningSystem (GPS) StandardPositioning Service (SPS) Perfor-
mance for 2018,” ARL Austin Texas,US government, March 2019.
4. Kaplan E. & egarty C., “Understanding GPS Principles and Applications”, Artech
House, 2006, 2nd Ed., 730 p.
5. G. Farolfi, S. Bianchini, N. Casagli, “Integration of GNSS and Satellite InSAR Data: Deri-
vation of Fine-Scale Vertical Surface Motion Maps of Po Plain, Northern Apennines, and
Southern Alps, Italy,” IEEE Transactions on Geoscience and Remote Sensing, vol. 57, pp
319-328, 2019.
6. . Leick, L. Rapoport and D. Tatarnikov, “GPS Satellite Surveying,” Wiley, 2015, 4 th Ed.
Chapter 7, 810 p.
7. X. Gu, A. Lipp, "DGPS positioning using carrier phase for precision navigation", Position
Location and Navigation Symposium 1994. IEEE. IEEE, pp. 410-417, 2002.
8. .-M. Codol, M. Poncelet, A. Monin, M. Devy, "Safety robotic lawnmower with precise and
low-cost L1-only RTK-GPS positioning", Proc. IROS Workshop Perception Nav. Auton.
Veh. Human Environ., pp. 69-72, 2011.
9. B. Hofmann-Wellenhof, H. Lichtenegger, E. Wasle, "GNSS - Global Navigation Satellite
Systems GPS GLONASS Galileo and more" in , New York:Springer Wien, 2008.